I've come across the following simple, but unexpected equality numerous times accidentally.
$$ \frac{10!}{6!} = 7! $$
which is the same as
$$1*2*3*4*5*6*7 = 7*8*9*10$$
Does it hold any specific (geometric?) meaning? I've graphed out numerous similar identities, but didn't find any that have an unexpected result similar to the above (leaving aside the trivial $\frac{n!}{(n-1)!} = n$).
If you're looking for some explanation as to why the two factorial expressions are equal, just break each expression down into prime factors. $$1*2*3*4*5*6*7=7*8*9*10$$
$$1*2*3*(2^2)*5*(2*3)*7=7*(2^3)*(3^2)*(2*5)$$
$$1*2^4*3^2*5*7=1*2^4*3^2*5*7$$
I know this only scratches at the surface of your questions about it, but I hope you still find it of some value.